The aerodynamics of flying snake airfoils in tandem configuration.

Flying snakes flatten their body to form a roughly triangular cross-sectional shape, enabling lift production and horizontal acceleration. While gliding, they also assume an S-shaped posture, which could promote aerodynamic interactions between the fore and the aft body. Such interactions have been studied experimentally; however, very coarse models of the snake's cross-sectional shape were used, and the effects were measured only for the downstream model. In this study, the aerodynamic interactions resulting from the snake's posture were approximated using two-dimensional anatomically accurate airfoils positioned in tandem to mimic the snake's geometry during flight. Load cells were used to measure the lift and drag forces, and flow field data were obtained using digital particle image velocimetry (DPIV). The results showed a strong dependence of the aerodynamic performance on the tandem arrangement, with the lift coefficients being generally more influenced than the drag coefficients. Flow field data revealed that the tandem arrangement modified the separated flow and the wake size, and enhanced the lift in cases in which the wake vortices formed closer to the models, producing suction on the dorsal surface. The downforce created by the flow separation from the ventral surface of the models at 0 deg angle of attack was another significant factor contributing to lift production. A number of cases showing large variations of aerodynamic performance included configurations close to the most probable posture of airborne flying snakes, suggesting that small postural variations could be used to control the glide trajectory.

Control of gliding in a flying snake-inspired n-chain model.

Flying snakes of genus Chrysopelea possess a highly dynamic gliding behavior, which is dominated by an undulation in the form of lateral waves sent posteriorly down the body. The resulting high-amplitude periodic variations in the distribution of mass and aerodynamic forces have been hypothesized to contribute to the stability of the snake's gliding trajectory. However, a previous 2D analysis in the longitudinal plane failed to reveal a significant effect of undulation on the stability in the pitch direction. In this study, a theoretical model was used to examine the dynamics and stability characteristics of flying snakes in three dimensions. The snake was modeled as an articulated chain of airfoils connected with revolute joints. Along the lines of vibrational control methods, which employ high-amplitude periodic inputs to produce desirable stable motions in nonlinear systems, undulation was considered as a periodic input to the system. This was implemented either by directly prescribing the joint angles as periodic functions of time (kinematic undulation), or by assuming periodic torques acting at the joints (torque undulation). The aerodynamic forces were modeled using blade element theory and previously determined force coefficients. The results show that torque undulation, along with linearization-based closed-loop control, could increase the size of the basin of stability. The effectiveness of the stabilization provided by torque undulation is a function of the amplitude and frequency of the input. In addition, kinematic undulation provides open-loop stability for sufficiently large frequencies. The results suggest that the snakes need some amount of closed-loop control despite the clear contribution of undulation to glide stability. However, as the closed-loop control system needs to work around a passively stable trajectory, undulation lowers the demand for a complex closed-loop control system. Overall, this study demonstrates the possibility of maintaining stability during gliding using a morphing body instead of symmetrically paired wings.

A theoretical analysis of pitch stability during gliding in flying snakes.

Flying snakes use their entire body as a continuously morphing 'wing' to produce lift and shallow their glide trajectory. Their dominant behavior during gliding is aerial undulation, in which lateral waves are sent posteriorly down the body. This highly dynamic behavior, which is unique among animal gliders, should have substantial effects on the flight dynamics and stability of the snakes, resulting from the continuous redistribution of mass and aerodynamic forces. In this study, we develop two-dimensional theoretical models to assess the stability characteristics of snakes in the pitch direction. Previously measured force coefficients are used to simulate aerodynamic forces acting on the models, and undulation is simulated by varying mass. Model 1 is a simple three-airfoil representation of the snake's body that possesses a passively stable equilibrium solution, whose basin of stability contains initial conditions observed in experimental gliding trajectories. Model 2 is more sophisticated, with more degrees of freedom allowing for postural changes to better represent the snake's real kinematics; in addition, a restoring moment is added to simulate potential active control. The application of static and dynamic stability criteria show that Model 2 is passively unstable, but can be stabilized with a restoring moment. Overall, these models suggest that undulation does not contribute to stability in pitch, and that flying snakes require a closed-loop control system formed around a passively stable dynamical framework.

Non-equilibrium trajectory dynamics and the kinematics of gliding in a flying snake.

Given sufficient space, it is possible for gliding animals to reach an equilibrium state with no net forces acting on the body. In contrast, every gliding trajectory must begin with a non-steady component, and the relative importance of this phase is not well understood. Of any terrestrial animal glider, snakes exhibit the greatest active movements, which may affect their trajectory dynamics. Our primary aim was to determine the characteristics of snake gliding during the transition to equilibrium, quantifying changes in velocity, acceleration, and body orientation in the late phase of a glide sequence. We launched 'flying' snakes (Chrysopelea paradisi) from a 15 m tower and recorded the mid-to-end portion of trajectories with four videocameras to reconstruct the snake's body position with mm to cm accuracy. Additionally, we developed a simple analytical model of gliding assuming only steady-state forces of lift, drag and weight acting on the body and used it to explore effects of wing loading, lift-to-drag ratio, and initial velocity on trajectory dynamics. Despite the vertical space provided to transition to steady-state gliding, snakes did not exhibit equilibrium gliding and in fact displayed a net positive acceleration in the vertical axis, an effect also predicted by the analytical model.